| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 7-s − 8-s + 9-s − 11-s − 2·12-s + 14-s + 16-s + 17-s − 18-s − 4·19-s + 2·21-s + 22-s + 23-s + 2·24-s + 4·27-s − 28-s + 2·29-s − 6·31-s − 32-s + 2·33-s − 34-s + 36-s + 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.577·12-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.436·21-s + 0.213·22-s + 0.208·23-s + 0.408·24-s + 0.769·27-s − 0.188·28-s + 0.371·29-s − 1.07·31-s − 0.176·32-s + 0.348·33-s − 0.171·34-s + 1/6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9690255361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9690255361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.07221212323098, −12.90075585778608, −12.21574482826442, −11.99826843552345, −11.23587887377890, −11.03780328044165, −10.56371017636009, −10.12455687477402, −9.592387998672326, −9.079765625075680, −8.530981510311880, −8.114110358992137, −7.350254664135574, −7.049261983294166, −6.467368809757905, −5.892189219032102, −5.655992458537266, −5.038579909281491, −4.300248763664615, −3.858915151351323, −2.941601598023168, −2.485608349099621, −1.768581901295592, −0.8050930362849369, −0.4956579258149277,
0.4956579258149277, 0.8050930362849369, 1.768581901295592, 2.485608349099621, 2.941601598023168, 3.858915151351323, 4.300248763664615, 5.038579909281491, 5.655992458537266, 5.892189219032102, 6.467368809757905, 7.049261983294166, 7.350254664135574, 8.114110358992137, 8.530981510311880, 9.079765625075680, 9.592387998672326, 10.12455687477402, 10.56371017636009, 11.03780328044165, 11.23587887377890, 11.99826843552345, 12.21574482826442, 12.90075585778608, 13.07221212323098
